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We show that the Cauchy problem for the sixth-order Boussinesq equation with cubic nonlinearity is globally well-posed in H s ( R ) provided 3 / 2 s 2 .

We consider the initial value problem associated to the nonlinear periodic sixth-order Boussinesq equation. For given initial data in Sobolev spaces H s ( T ) , the local well-posedness of this equation is established for s > − 1 / 2 .

In this paper we prove local well-posedness in L 2 ( R ) and H 1 ( R ) for the generalized sixth-order Boussinesq equation u t t = u x x + β u x x x x + u x x x x x x + ( | u | α u ) x x . Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) [14] . We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.